How random matrices can elucidate mysteries of number theory: The Cohen-Lenstra-Martinet heuristics, roots of unity, and random matrix theory

Abstract:Class groups of number field extensions are objects of fundamental importance in algebraic number theory, since they measure the failure of unique factorization in the rings of integers of these extensions. These groups are finite and abelian, but, even after hundreds of years of study, very little else is known about their structure in general.

However, the Cohen-Lenstra-Martinet heuristics yield a conjectural frequency with which a fixed finite abelian group appears as an ideal class group of an extension of number fields, for certain sets of extensions of a fixed base field. Recently, Malle found numerical evidence suggesting that their proposed frequency is incorrect when there are unexpected roots of unity in the base field of these extensions. Moreover, Malle proposed a new frequency, which is a much better match for his data.

I will explain a random matrix heuristic (coming from function fields) that leads to a function field version of Malle's conjecture (as well as generalizations of it).

Bio:Derek Garton is a Postdoctoral Lecturer at the Department of Mathematics of Northwestern University. He researches number theory, arithmetic geometry, and arithmetic dynamics. Derek grew up in Hillsboro, Oregon. After graduating from Whitman College, he earned a Maîtrise de Mathématiques from the Université Pierre et Marie Curie (Paris) and a MS in mathematics from Portland State University. In 2012, Derek earned his PhD from the University of Wisconsin under the direction of Jordan Ellenberg. It was in Wisconsin that Derek discovered his favorite kind of theory was number theory. His favorite food cart is the Grilled Cheese Grill and his favorite Trail Blazer is Wesley Matthews.

Monday, February 11th, 2013 at 3:15PMNeuberger Hall room 454This event is free and open to the public.